Matrix realization of random surfaces

The large-N one-matrix model with a potential V()=22+g44N+g66N2 is carefully investigated using the orthogonal polynomial method. We present a numerical method to solve the recurrence relation and evaluate the recursion coefficients rk (k=1, 2, 3, ) of the orthogonal polynomials at large N. We find that for g6g42>12 there is no m=2 solution which can be expressed as a smooth function of kN in the limit N. This means that the assumption of smoothness of rk at N near the critical point, which was essential to derive the string susceptibility and the string equation, is broken even at the tree level of the genus expansion by adding the 6 term. We have also observed the free energy around the (expected) critical point to confirm that the system does not have the desired criticality as pure gravity. Our (discouraging) results for m=2 are complementary to previous analyses by the saddle-point method. On the other hand, for the case m=3 (g6g42=45), we find a well-behaved solution which coincides with the result obtained by Brézin, Marinari, and Parisi. To strengthen the validity of our numerical scheme, we present in an appendix a nonperturbative solution for m=1 which obeys the so-called type-II string equation.